Deformative Magnetic Marked Length Spectrum Rigidity

Abstract

Let $M$ be a closed surface and let $\{g_s \ | \ s \in (-\epsilon, \epsilon)\}$ be a smooth one-parameter family of Riemannian metrics on $M$. Also let $\{\kappa_s : M \rightarrow \mathbb{R} \ | \ s \in (-\epsilon, \epsilon)\}$ be a smooth one-parameter family of functions on $M$. Then the family $\{(g_s, \kappa_s) \ | \ s \in (-\epsilon, \epsilon)\}$ gives rise to a family of magnetic flows on $TM$. We show that if the magnetic curvatures are negative for $s \in (-\epsilon, \epsilon)$ and the lengths of each periodic orbit remains constant as the parameter $s$ varies, then there exists a smooth family of diffeomorphisms $\{f_s : M \rightarrow M \ | \ s \in (-\epsilon, \epsilon)\}$ such that $f_s^*(g_s) = g_0$ and $f_s^*(\kappa_s) = \kappa_0$. This generalizes a result of Guillemin and Kazhdan to the setting of magnetic flows.